cp's OEIS Frontend

A348617 Numbers whose sum of prime indices is twice their negated alternating sum.

%I A348617 #14 Dec 10 2021 11:11:50
%S A348617 1,10,39,88,115,228,259,306,517,544,620,783,793,870,1150,1204,1241,
%T A348617 1392,1656,1691,1722,1845,2369,2590,2596,2775,2944,3038,3277,3280,
%U A348617 3339,3498,3692,3996,4247,4440,4935,5022,5170,5226,5587,5644,5875,5936,6200,6321
%N A348617 Numbers whose sum of prime indices is twice their negated alternating sum.
%C A348617 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A348617 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C A348617 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are also Heinz numbers of partitions whose sum is twice their negated alternating sum.
%F A348617 A056239(a(n)) = -2*A316524(a(n)).
%F A348617 A346698(a(n)) = 3*A346697(a(n)).
%e A348617 The terms and their prime indices begin:
%e A348617      1: ()
%e A348617     10: (3,1)
%e A348617     39: (6,2)
%e A348617     88: (5,1,1,1)
%e A348617    115: (9,3)
%e A348617    228: (8,2,1,1)
%e A348617    259: (12,4)
%e A348617    306: (7,2,2,1)
%e A348617    517: (15,5)
%e A348617    544: (7,1,1,1,1,1)
%e A348617    620: (11,3,1,1)
%e A348617    783: (10,2,2,2)
%e A348617    793: (18,6)
%e A348617    870: (10,3,2,1)
%e A348617   1150: (9,3,3,1)
%e A348617   1204: (14,4,1,1)
%e A348617   1241: (21,7)
%e A348617   1392: (10,2,1,1,1,1)
%e A348617   1656: (9,2,2,1,1,1)
%e A348617   1691: (24,8)
%t A348617 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A348617 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A348617 Select[Range[1000],Total[primeMS[#]]==-2*ats[primeMS[#]]&]
%Y A348617 These partitions are counted by A001523 up to 0's.
%Y A348617 An ordered version is A349154, nonnegative A348614, reverse A349155.
%Y A348617 The nonnegative version is A349159, counted by A000712 up to 0's.
%Y A348617 The reverse nonnegative version is A349160, counted by A006330 up to 0's.
%Y A348617 A027193 counts partitions with rev-alt sum > 0, ranked by A026424.
%Y A348617 A034871, A097805, A345197 count compositions by alternating sum.
%Y A348617 A035363 = partitions with alt sum 0, ranked by A066207, complement A086543.
%Y A348617 A056239 adds up prime indices, row sums of A112798, row lengths A001222.
%Y A348617 A103919 counts partitions by alternating sum, reverse A344612.
%Y A348617 A344607 counts partitions with rev-alt sum >= 0, ranked by A344609.
%Y A348617 A346697 adds up odd-indexed prime indices.
%Y A348617 A346698 adds up even-indexed prime indices.
%Y A348617 Cf. A000984, A001700, A028260, A045931, A120452, A195017, A257991, A257992, A262977, A325698, A344619, A345958.
%K A348617 nonn
%O A348617 1,2
%A A348617 _Gus Wiseman_, Nov 26 2021